Multiplying Polynomials - FOIL Method

When we are exploring polynomial multiplication; the FOIL method of polynomials can't be ignored or can't be taken lightly. Grade 8 or higher students must know how to multiply two binomials and "FOIL" method is a good method to learn to multiply two binomials. As usual, we start our journey from the basics and I want to decode the word "FOIL", first of all.

This word is decoded as below; "F" means; Firsts. "O" means; Outers. "I" means; Inners and "L" means; Lasts. Now, let's move further to understand, what are "Firsts", "Outers", "Inners" and "Lasts"? Consider two binomials "2x - 3"and "3a + 4"and we want to multiply both of these binomials. To multiply these binomials we need to write them as shown below to start the solution for the multiplication: (2x - 3) (3a + 4) As we used the brackets (parenthesis), we need not to show multiply sign between them because if there is nothing between two brackets it means they are getting multiplied with each other.
Now, look at both binomials in the brackets, "2x" and "3a" are called "Firsts".
"2x" and "4" are called "Outers".

"- 3" and "3a" are called "Inners".

"- 3" and "4" are called "Lasts".

All the above four code words together make the code word "FOIL" when first letter is chosen from each.

Solve the given binomial product by using "FOIL": Rewrite both the binomials as shown below: (2x - 3) (3a + 4) Now, multiply the firsts, which are "2x" and "3a" to get "6ax" as the first term of the solution. (Read my previous article if you don't know how to get "6ax" from "2x" and "3a") Multiply the outers, which are "2x" and "4" to get "8x" as the second term of the solution.

Multiply the inners, which are "- 3" and "3a" to get "- 9a" as the third term of the solution.

Finally, multiply the lasts, which are "- 3" and "4" to get "- 12".
Let's rewrite our binomials and all the terms in the solutions the way it should be shown in your tests, quizzes or homework.

Solution: (2x - 3) (3a + 4) = 6ax + 8x - 9a - 12 All the four terms are different and we can't combine any of them, hence above is our final answer for the binomial multiplication.

Let's do another example to understand the concept further and having similar terms in the first step of the solution.

Multiply the following binomials (Some questions just ask simplify or others can ask find the product, but all meaning same) (5n - 2) (-2n + 3) Solution: Rewrite the both monomials and "FOIL" them as shown below: (5n - 2) (-2n + 3) = -10n²+15n + 4n - 6 = -10n²+ 19n - 6 Explanation of the first step in the solution: Notice that we multiply the "firsts" as; (5n) (- 2n) = - 10n² We got "+15n" by multiplying the "outers" as; (5n) (3) = 15n We got "4n" by multiplying the "inners" as; (- 2) (- 2n) = 4n We got "- 6" by multiplying "lasts" as; (- 2) (3) = - 6 Explanation of the second step: After foiling the binomials in the first step, we have four terms (always in case of binomial multiplication).
Now look for any similar terms.
Did you find any similar terms in the first step? I did.
Terms "+ 15n" and "+ 4n" are similar. Why? You should know it, if you are reading all my articles. Hence combine "+ 15n" and "+ 4n" to get "+ 19n".
Now we have all three different terms in our solution, and hence is our answer.
That's what I am teaching to my math students at my tuition center about "FOIL". But there is another cool method to multiply binomials and many of my students like that method as well.

I will explain that method at my site in near future.
To learn more algebra, stay tuned.